Fubini's nightmare

Fubini's nightmare is a seeming violation of Fubini's theorem, where a nice space, such as the square is foliated by smooth fibers, but there exists a set of positive measure whose intersection with each fiber is singular (at most a single point in Katok's example). There is no real contradiction to Fubini's theorem because despite smoothness of the fibers, the foliation is not absolutely continuous, and neither are the conditional measures on fibers.

Existence of Fubini's nightmare complicates fiber-wise proofs for center foliations of partially hyperbolic dynamical systems: these foliations are typically Hölder but not absolutely continuous.

A hands-on example of Fubuni's nightmare was suggested by Anatole Katok and published by John Milnor.[1] A dynamical version for center foliation was constructed by Amie Wilkinson and Michael Shub.[2]

Katok's construction

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Foliation

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For a consider the coding of points of the interval by sequences of zeros and ones, similar to the binary coding, but splitting the intervals in the ratio . (As for the binary coding, we identify with )

The point, corresponding to a sequence is given explicitly by

where is the length of the interval after first splits.

Katok's foliation

For a fixed sequence the map is analytic. This follows from the Weierstrass M-test: the series for converges uniformly on compact subsets of the intersection In particular, is an analytic curve.

Now, the square is foliated by analytic curves

Set

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For a fixed and random sampled according to the Lebesgue measure, the coding digits are independent Bernoulli random variables with parameter , namely and

By the law of large numbers, for each and almost every

By Fubini's theorem, the set

has full Lebesgue measure in the square .

However, for each fixed sequence the limit of its Cesàro averages is unique, if it exists. Thus every curve either does not intersect at all (if there is no limit), or intersects it at the single point where

Therefore, for the above foliation and set , we observe a Fubini's nightmare.

Wilkinson–Shub construction

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Wilkinson and Shub considered diffeomorphisms which are small perturbations of the diffeomorphism of the three dimensional torus where  is the Arnold's cat map. This map and its small perturbations are partially hyperbolic. Moreover, the center fibers of the perturbed maps are smooth circles, close to those for the original map.

The Wilkinson and Shub perturbation is designed to preserve the Lebesgue measure and to make the diffeomorphism ergodic with the central Lyapunov exponent Suppose that is positive (otherwise invert the map). Then the set of points, for which the central Lyapunov exponent is positive, has full Lebesgue measure in

On the other hand, the length of the circles of the central foliation is bounded above. Therefore, on each circle, the set of points with positive central Lyapunov exponent has to have zero measure. More delicate arguments show that this set is finite, and we have the Fubini's nightmare.

References

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  1. ^ Milnor, J. (1997). "Fubini Foiled: Katok's Paradoxical Example in Measure Theory". The Mathematical Intelligencer. 19 (2): 30–32. doi:10.1007/BF03024428.
  2. ^ Shub, M.; Wilkinson, A. (2000). "Pathological foliations and removable zero exponents". Inventiones mathematicae. 139: 495–508. doi:10.1007/s002229900035.